PHYSICAL REVIEW A VOLUME 49, NUMBER 2 FEBRUARY 1994 Hyperspherical functions with arbitrary permutational symmetry Akiva Novoselsky Racah Institute of Physics, The Hebrew University, Jerusalem 9190$, Israel Jacob Katriel Department of Chemistry, Technion Isra— el Institute of Technology, Haifa OOOO, Israel (Received 29 July 1993) An algorithm is formulated for the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates. A recursive procedure is proposed, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the hscfps calculated in the preceding step as well as the Raynal-Revai and the T coefBcients. The results are applicable to the study of the atomic, molecular, and nuclear few-body problem. PACS number(s): 03. 65.Ge, 02.2Q. Df, 36. 40.+d, 21. 45.+v I. INTRODUCTION The method of hyperspherical coordinates, along with the associated hyperspherical functions, was introduced in 1935 by Zernike and Brinkman [1]. Delves [2) and Smith [3] reintroduced this method in a different form 25 years later, and it was recently reviewed by Nikiforov et al. [4]. The hyperspherical functions were extensively used in recent years to study few-body problems in nu- clear, atomic, and molecular physics [5]. In this method the internal degrees of freedom of an N- body system are reduced to a single hyperradial coordi- nate and a set of 3N — 4 angular coordinates. The hyper- radius is invariant under particle permutations, making the hyperspherical coordinates very useful for rearrange- ment processes. A further convenient feature is that each hyperspherical basis function is separable into a product of a function of the hyperradius and a function of the hyperangular coordinates. Since the total (space-spin) wave functions of a many- body system should be antisymmetric, the construction of permutational symmetry adapted hyperspherical func- tions has been considered extensively. Nevertheless, no generally valid efficient procedure, which is applicable for systems consisting of more than three particles, has been developed so far [6]. Four years ago, we introduced a recursive procedure for the construction of nonspurious harmonic oscilla- tor functions with arbitrary permutational symmetry, in Jacobi coordinates [7]. According to this procedure, each symmetry-adapted N-particle harmonic oscillator function is written as a linear combination of angular- momentum coupled products of permutational symme- try adapted (N — 1)-particle wave functions and an Nth- particle wave function, all members of the linear com- bination having the same N-particle harmonic oscillator energy. The coefficients of this linear combination are the harmonic oscillator coefficients of &actional parentage. These coefficients are the eigenvectors of the transposi- tion class sum of the symmetric group, in the appropriate basis. In the present article we use a variant of this recursive method to evaluate hyperspherical functions belonging to well defined irreducible representations (irreps) of the symmetric group. We diagonalize the transposition class sum of the symmetric group within invariant subspaces with respect to S~, spanned by appropriate hyperspher- ical functions. Each of these functions is an angular mo- mentum as well as hyperspherical angular momentum coupled product of an (N — 1)-particle permutational symmetry adapted hyperspherical function with a sin- gle particle function. The eigenvalues that are obtained after the diagonalization uniquely identify the irreps of the symmetric group. The eigenvectors are the hyper- spherical coefficients of &actional parentage (hscfps). In the actual computation only the matrix element of the transposition of the last two particles has to be evaluated. The presentation is organized as follows: In Secs. II and III we introduce the hyperspherical coordinates and the hyperspherical Laplacian, respectively. The hyper- spherical functions are presented in Sec. IV. In Sec. V we brie6y review the Raynal-Revai and the T coefficients that are used in the subsequent sections. In Sec. VI we summarize the relevant part of the representation theory of the symmetric group. The permutational symmetry adapted hyperspherical basis is presented in Sec. VII for three particles and in Sec. VIII for N particles. The computational algorithm is given in Sec. IX. Some con- cluding remarks are made in Sec. X. II. THE HYPERSPHERICAL COORDINATES Several variants of hyperspherical coordinates have been used by different authors. They differ by both the choice of the underlying set of "single-particle, " co- 1050-2947/94/49(2)/833(14)/$06. 00 49 833 1994 The American Physical Society